Conditional Extropy: Uncertainty Functionals

• Conditional extropy is a family of uncertainty measures, defined as the complementary dual of entropy, that localize information through diverse conditioning methods.
• It encompasses formulations for discrete variables, interval-conditioned differential measures, weighted cumulative past extropy, and dynamic reliability applications.
• The theory highlights non-additivity and monotonicity properties while adapting extropy measures to various probabilistic contexts with bounded and unbounded supports.

Conditional extropy is a family of extropy-based uncertainty functionals evaluated after conditioning on information. In the current arXiv literature, the term covers several technically distinct constructions: conditional extropy for discrete random variables defined through conditional pmfs, interval-conditioned differential extropy for truncated continuous laws, conditional weighted cumulative past extropy defined relative to a sub-$\sigma$-field, and dynamic reliability measures built from conditional CDF or survival ratios for residual or past life. Across these formulations, extropy is treated as the complementary dual of entropy, but the conditional theory is not unified by a single chain rule; instead, it appears as a collection of conditionally defined quadratic functionals adapted to discrete information, interval truncation, or lifetime conditioning (Kumar et al., 15 Jul 2025, Gupta et al., 2022, Sahu et al., 2022, Chaudhary et al., 16 Apr 2026, Pandey et al., 2024).

1. Foundational setting and scope

For a continuous nonnegative random variable $X$ with pdf $f$, extropy is defined as

$J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$

while for a discrete random variable with pmf $\{p_i\}$, extropy is

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$

The literature cited here consistently describes extropy as the complementary dual of entropy. It also treats cumulative and weighted analogues as part of the same hierarchy, including cumulative past extropy, weighted cumulative past extropy, and general weighted cumulative residual or past extropy for order statistics (Kumar et al., 15 Jul 2025, Sahu et al., 2022, Chaudhary et al., 16 Apr 2026).

Within that hierarchy, conditionalization is implemented in several different ways. One line conditions on another random variable through a conditional pmf; another conditions on an interval $S=\{c<Y<d\}$ by replacing the density with the truncated density; another replaces the unconditional CDF by a conditional CDF $F_X(x\mid\mathcal G)$ relative to a sub-$\sigma$-field; and reliability-theoretic work uses conditional survival or failure ratios such as $\overline F(x)/\overline F(t)$ or $X$0 to define dynamic residual or past extropies. This suggests that conditional extropy is best understood as a family of conditionally localized extropy measures rather than a single universal object (Gupta et al., 2022, Sahu et al., 2022, Chaudhary et al., 16 Apr 2026).

2. Interval-conditioned differential extropy

A direct continuous formulation is given for an absolutely continuous random variable $X$1 with density $X$2 and CDF $X$3. For the interval

$X$4

the conditional density is

$X$5

The corresponding conditional extropy is defined by

$X$6

or equivalently

$X$7

The same paper recalls ordinary differential extropy as

$X$8

so $X$9 is the interval-conditioned analogue obtained by truncation and renormalization (Gupta et al., 2022).

The central structural result is a partial monotonicity theorem. If $f$0 is log-concave, then $f$1 is increasing in $f$2 for fixed $f$3; if $f$4 is log-convex, then $f$5 is decreasing in $f$6. The proof differentiates

$f$7

with respect to $f$8, introduces the auxiliary function

$f$9

uses $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$0, and derives the sign from the log-concavity or log-convexity of $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$1. Under log-concavity, the argument yields

$J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$2

The authors interpret this as uncertainty increasing when the conditioning interval expands by increasing $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$3, because a larger interval corresponds to less precise information (Gupta et al., 2022).

The same work places the result beside other extropy variants. It recalls weighted extropy

$J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$4

and general weighted extropy

$J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$5

with conditional form

$J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$6

Although the interval-monotonicity theorem is for the unweighted case, the paper’s convolution section uses the same mechanism in a weighted setting. Log-concave Weibull densities with $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$7 and Gamma densities with shape parameter $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$8 are given as examples illustrating how log-concavity and interval truncation lead to monotone extropy behavior (Gupta et al., 2022).

3. Discrete conditional extropy and non-additivity

For discrete random variables $J(X)=-\frac12\int_0^\infty f^2(x)\,dx,$9 and $\{p_i\}$0 with joint pmf $\{p_i\}$1, marginal pmf $\{p_i\}$2 for $\{p_i\}$3, and conditional pmf $\{p_i\}$4, conditional extropy is defined in two steps: $\{p_i\}$5 and

$\{p_i\}$6

This is the discrete extropy analogue of conditional entropy, but with the kernel $\{p_i\}$7 rather than $\{p_i\}$8. The same paper emphasizes the complementary identity

$\{p_i\}$9

which formalizes the entropy-extropy duality at the one-variable level (Kumar et al., 15 Jul 2025).

A defining feature of the discrete theory is non-additivity. For joint extropy

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$0

the paper explicitly notes that, unlike entropy,

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$1

extropy does not generally satisfy

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$2

Its numerical example gives

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$3

so the would-be chain rule fails. This is one of the main reasons conditional extropy is introduced as an analytic tool rather than as one component of a classical additive decomposition (Kumar et al., 15 Jul 2025).

The paper nonetheless proves several properties usually expected of a conditional uncertainty measure. If $J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$4 and $J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$5 are independent, then

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$6

It also states the strict uncertainty-reduction relation

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$7

Further, under the condition

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$8

it proves

$J(X)=-\sum_{i=1}^{n}(1-p_i)\log(1-p_i).$9

where $S=\{c<Y<d\}$0 and $S=\{c<Y<d\}$1 are the support sizes. Under the opposite regime,

$S=\{c<Y<d\}$2

the reverse bound is obtained: $S=\{c<Y<d\}$3 The sign change is tied to the derivative of the basic kernel $S=\{c<Y<d\}$4, which changes sign at $S=\{c<Y<d\}$5 (Kumar et al., 15 Jul 2025).

The same work also gives

$S=\{c<Y<d\}$6

linking conditional extropy to a generalized conditional extropy term and to Shannon entropy. In addition, for distributions in $S=\{c<Y<d\}$7 with probabilities bounded by $S=\{c<Y<d\}$8, extropy is Lipschitz continuous under the $S=\{c<Y<d\}$9-metric: $F_X(x\mid\mathcal G)$0 The authors use conditional extropy as part of the conceptual bridge to extropy rate for stochastic processes, precisely because naive average joint extropy per variable does not provide an entropy-style rate theory (Kumar et al., 15 Jul 2025).

4. Conditional weighted cumulative past extropy

A separate branch of the literature studies conditional weighted cumulative past extropy, abbreviated CWCPJ. For a nonnegative absolutely continuous random variable $F_X(x\mid\mathcal G)$1 with bounded support $F_X(x\mid\mathcal G)$2, and a sub-$F_X(x\mid\mathcal G)$3-field $F_X(x\mid\mathcal G)$4, the conditional weighted cumulative past extropy is

$F_X(x\mid\mathcal G)$5

Here $F_X(x\mid\mathcal G)$6 is the conditional CDF, written in the paper as the conditional expectation of the indicator $F_X(x\mid\mathcal G)$7. The bounded-support assumption is essential: the paper repeatedly states that weighted cumulative past extropy and its conditional version are only meaningful on bounded support because otherwise the integral diverges to $F_X(x\mid\mathcal G)$8 (Sahu et al., 2022).

CWCPJ generalizes the unconditional weighted cumulative past extropy

$F_X(x\mid\mathcal G)$9

If $\sigma$0 is a trivial $\sigma$1-field, then

$\sigma$2

The paper also proves a Jensen-type conditioning inequality: if $\sigma$3 for some $\sigma$4 and $\sigma$5, then

$\sigma$6

This is presented as a monotonicity effect under refinement of the conditioning $\sigma$7-field (Sahu et al., 2022).

A further theorem gives an upper bound in terms of conditional extropy: $\sigma$8 where

$\sigma$9

The paper describes this as the conditional analogue of its earlier unconditional extropy bound. Another major result is

$\overline F(x)/\overline F(t)$0

with equality if and only if $\overline F(x)/\overline F(t)$1 is independent of $\overline F(x)/\overline F(t)$2. This provides an exact independence characterization in extropy terms (Sahu et al., 2022).

The Markov structure is also explicit. If

$\overline F(x)/\overline F(t)$3

is a Markov chain, then

$\overline F(x)/\overline F(t)$4

and

$\overline F(x)/\overline F(t)$5

The paper places these results in the hierarchy

$\overline F(x)/\overline F(t)$6

thereby making the conditional object a formal extension of earlier cumulative extropy measures (Sahu et al., 2022).

5. Dynamic and reliability-theoretic conditional forms

In reliability theory, dynamic extropy measures are defined by conditioning on survival beyond $\overline F(x)/\overline F(t)$7 or failure before $\overline F(x)/\overline F(t)$8. For a nonnegative absolutely continuous lifetime $\overline F(x)/\overline F(t)$9, the dynamic weighted cumulative residual extropy is

$X$00

and the dynamic weighted cumulative past extropy is

$X$01

For extreme order statistics,

$X$02

and

$X$03

The paper explicitly says these dynamic measures are conditional extropy measures in the sense of reliability and information theory, but not in the strict Shannon-style conditional information sense; they are conditionalized or truncated extropy functionals based on the ratios $X$04 or $X$05 (Chaudhary et al., 16 Apr 2026).

The same framework yields monotonicity and characterization results for order statistics. For the minimum,

$X$06

is increasing in $X$07 when $X$08, and satisfies both

$X$09

and

$X$10

For the maximum,

$X$11

is non-decreasing in $X$12, satisfies

$X$13

and has the lower bound

$X$14

The paper further states that these weighted measures and their dynamic counterparts uniquely characterize the underlying distribution, with specific characterizations for the generalized Pareto distribution and the power distribution (Chaudhary et al., 16 Apr 2026).

A genuinely multivariate development appears in the study of bivariate dynamic conditional failure extropy. For an absolutely continuous non-negative random vector $X$15 with distribution function $X$16, the conditional dynamic cumulative failure extropy components are

$X$17

and

$X$18

These quantify uncertainty in the conditional past lifetime distributions of $X$19 and $X$20 given $X$21. The paper links them to the bivariate reversed hazard rate and the expected inactivity time vector, and proves the expectation representation

$X$22

It also establishes the lower bound

$X$23

derives an entropy-type upper bound, and states that CCDFEx uniquely determines the joint distribution function (Pandey et al., 2024).

The bivariate paper extends the theory to stochastic ordering, proportional reversed hazard models, and estimation. It defines a partial order $X$24, shows that ordering of marginal survival variables or of reversed hazard components induces CCDFEx ordering, and provides both empirical plug-in and kernel-smoothed estimators. Under a non-increasing bandwidth sequence $X$25 with $X$26, the paper reports consistency and weak convergence via the Glivenko–Cantelli theorem. In simulations from a bivariate exponential distribution with correlation coefficient $X$27, mean vector $X$28, and sample sizes $X$29, the kernel estimator has consistently smaller average bias and MSE than the empirical estimator (Pandey et al., 2024).

6. Structural themes, limitations, and adjacent developments

Several recurring principles emerge across these formulations. First, conditioning frequently reduces extropy-like uncertainty in an averaged or information-refined sense: $X$30 in the discrete setting; $X$31 for CWCPJ; and the dynamic lifetime literature explicitly notes that conditioning has a decreasing effect on cumulative past extropy. Second, monotonicity need not take the same form everywhere: interval-conditioned differential extropy can increase with the upper truncation point $X$32 under log-concavity, because enlarging the admissible interval represents less precise information (Kumar et al., 15 Jul 2025, Sahu et al., 2022, Chaudhary et al., 16 Apr 2026, Gupta et al., 2022).

A common misconception is that conditional extropy should parallel conditional entropy in all formal respects. The discrete theory shows that this is false at the level of joint decomposition, since $X$33 in general. The dynamic order-statistic paper makes a related point from another direction: its residual and past versions are conditional extropy measures for truncated laws, but not conditional entropy in the classical chain-rule sense. These distinctions are central rather than peripheral, because they explain why the literature develops multiple conditional extropy notions tailored to distinct probabilistic structures (Kumar et al., 15 Jul 2025, Chaudhary et al., 16 Apr 2026).

Another limitation is support dependence. Weighted cumulative past extropy and its conditional version require bounded support, since otherwise the corresponding integrals become $X$34. By contrast, differential and survival-based extropy constructions are formulated directly on unbounded supports when the defining integrals exist. The literature therefore separates conditional extropy into support-sensitive subfamilies rather than offering a single general formula valid across discrete, absolutely continuous, bounded-support, and lifetime-truncation settings (Sahu et al., 2022, Chaudhary et al., 16 Apr 2026).

An adjacent development concerns extropy-based relative measures. One paper explicitly states that it does not define conditional extropy in the usual sense, but instead develops generalized extropy inaccuracy, the generalized extropy divergence ratio, and the generalized extropy similarity ratio. It also states that these quantities are naturally related to conditional extropy and can be specialized or extended to conditional settings by conditioning the underlying density, CDF, or survival functions. A plausible implication is that future work may connect conditional extropy more tightly to normalized relative-information geometry, since that paper proves cosine-squared representations, scale invariance, and model-based bounds for the corresponding extropy ratios (P. et al., 19 Aug 2025).